= Coulomb exchange splitting of two orbitals
{c}
{title2=$E_{\pm}=\epsilon_a+\epsilon_b+J\pm K$}
Let $a,b$ be distinct orthonormal spatial orbitals occupied by two identical <spin one-half> particles, with a weak repulsive <Coulomb interaction>. The symmetric and antisymmetric spatial combinations have first-order <energy> shifts $J+K$ and $J-K$, where $J$ is the direct Coulomb integral and
$$
K=g\int d^3r\,d^3r'\,\frac{a^*(\mathbf r)b(\mathbf r)b^*(\mathbf r')a(\mathbf r')}{|\mathbf r-\mathbf r'|},\qquad g>0.
$$
The Coulomb kernel has positive <Fourier transform> $4\pi/|\mathbf k|^2$, so $K\ge0$. <Fermionic exchange symmetry> pairs the symmetric spatial function with the <spin-one-half singlet state> and the antisymmetric one with the <spin-one-half triplet state>. For these fixed orbitals the triplet lies lower by $2K$ when $K>0$. This does not compare arbitrary singlets and triplets with different orbital occupations; for negligible orbital overlap the splitting can vanish.
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